Okay, so when I started making this list after finishing up Superman, I had no idea that whole thing about Hydra was going to happen. I'm...going to ignore it to be honest. If you're a new fan who's just getting into Cap, I suggest you do so as well. My guess is, in a few months, it'll be completely retconned anyway, because this is comics and everyone had a pretty negative reaction to it. For good reason.
For those of you still curious about Captain America, here's a list of places to start with, or story arcs in Captain America that are a good idea to know about. These are all trades, because I personally find reading heroes with as huge a back list as Cap goes much easier when read in trades.
If you want to start from the very very beginning, this series of trades is a good way to go. These collect a majority of the early issues of Captain America when the series first started back in the forties.
The seventies were an interesting time for Captain America. This series introduces Falcon and shows Cap becoming slightly disillusioned with America due to what's happening in the country (it's the seventies, it was a turbulent time for us).
If you're more interested in modern interpretations of Captain America, this series is a good place to start. It deals with the aftermath of 9/11 in the Marvel Universe.
Not to be confused with the series above, that was Steve and Sam meeting. In this one they already know each other, and Sam goes rogue (FYI, comics Sam is very different from MCU Sam).
If you've seen the second movie, you know why this is on here.
If you've seen the third movie, you know why this is on here. Just be warned: things go down very differently in the comics.
Yes, like Superman Cap died. Also like Superman, he came back. Comics.
Like I said, he comes back. Because, comics.
Bucky goes to trial and things from Cap's past come to light.
Looking to skip ahead and just catch up on what's happening with present day Cap? This is an excellent place to start.